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Chapter 3: Problem 55

Identify the slope of a line that is: perpendicular to the line \(y=8 x-11\).

Short Answer

Expert verified
The slope is \(- \frac{1}{8}\).

Step by step solution

01

Identify the slope of the given line

The equation of the line is given in slope-intercept form: \(y = 8x - 11\). The coefficient of \(x\) in this equation is the slope. Therefore, the slope of the given line is \(m_1 = 8\).
02

Understand the relationship between perpendicular lines

The slopes of two perpendicular lines in the plane are negative reciprocals of each other. This means if one line has a slope of \(m_1\), the slope of the perpendicular line \(m_2\) will be \(-\frac{1}{m_1}\).
03

Calculate the negative reciprocal

To find the slope of the line that is perpendicular to the given line, calculate the negative reciprocal of \(m_1 = 8\). Thus, \(m_2 = -\frac{1}{8}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope
The slope is a fundamental concept in algebra and geometry.
It measures the steepness or incline of a line.
In simple words, the slope tells you how much a line goes up or down as you move from left to right.
The slope is usually represented by the letter 'm'.
For a line with equation in the slope-intercept form, which we will discuss later, the slope is the coefficient of the variable 'x'. For example, in the equation

y = 8x - 11

the slope is 8. Signs of the slope have specific meanings:
  • A positive slope means the line is going up.

  • A negative slope means the line is going down.

  • If the slope is zero, the line is horizontal.

  • An undefined slope means the line is vertical.
The formula to calculate the slope between two points
(x_1, y_1) and (x_2, y_2) is given by:
\(m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\)
negative reciprocal
To understand perpendicular lines, you must know the concept of the negative reciprocal.
If you have a slope of a line as 'm', the negative reciprocal is found by flipping the number and changing the sign.
For example:
  • If the slope
slope-intercept form
The slope-intercept form is a simple and efficient way to express the equation of a straight line.
It is written as:
\(y = mx + b\)
where 'm' stands for slope and 'b' stands for the y-intercept (the point where the line crosses the y-axis).
For example, the equation:
  • \(y = 8x - 11\)
    has a slope (m) of 8 and a y-intercept (b) of -11.
This method makes it very easy to understand and graph a line.
By looking at the equation, you can immediately see how steep the line is and where it crosses the y-axis. This form is especially helpful when you need to:
  • Identify the slope quickly

  • Understand the y-intercept's position

  • Graph the line
    • . Learning the slope-intercept form is essential for solving many algebraic problems involving lines.

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Most popular questions from this chapter

According to the Pew Research Center, the Millennial Generation in the United States includes about 50 million people born after 1980 . Find the number of this generation who say that "you can't be too careful" when dealing with people. Round to the nearest tenth of a million. Whether as a by-product of protective parents, the age of terrorism or a media culture that focuses on dangers, they [Millennials] cast a wary eye on human nature. Two-thirds say "you can't be too careful" when dealing with people. (Source: www.pewresearch.org/millennials, 2010)

For exercises 67-78, (a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(4 x+9 y=72\)

(a) find three solutions of the equation. (b) graph the equation. \(y=-\frac{1}{4} x+3\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(\frac{1}{6}, 8\right) ;\left(\frac{5}{6}, 11\right)\)

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The amount of fresh tomatoes consumed per person in the United States increased from \(89.9\) lb in 2009 to 93.5 lb in 2011. (Source: www.ers.usda.gov, Dec. 15,2011\()\)
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